The only difference between Theorem REF  and Theorem REF  is that the arcs of \([X,Y]\)  must have unique labels in Theorem REF , whereas this is not required in Theorem REF .
To relax this requirement of Theorem REF , we require that any set of all arcs of \([X,Y]\)  with identical labels must arc-induce a complete bipartite graph.
By Lemma REF , these complete bipartite graphs are decomposable.
Then we have that all arcs of a complete bipartite subgraph \(B(X_1,Y_1), X_1\subseteq X, Y_1\subseteq Y,\)  of \(G\)  with the same label are synchronous arcs.
Furthermore, all other arcs of \(G\)  have labels different from the labels of \(B(X_1,Y_1)\) .
This means that \(B(X_1,Y_1)/Y_1G/X \subseteq G/YG/X\)  and \(G/Y B(X_1,Y_1)/X_1 \subseteq G/YG/X\) , and, due to Lemma REF , \(B(X_1,Y_1)/Y_1B(X_1,Y_1)/X_1\cong B(X_1,Y_1)\) .
Therefore, \(G\cong G/YG/X\) , which we prove in Theorem REF .
In Theorem REF , we use the proof of Theorem REF  given in [1]} and modify this proof to support complete bipartite subgraph of \(G\)  with arcs with the same label of \([X,Y]\) .
